JournalsjemsVol. 14, No. 6pp. 2001–2038

The Kähler Ricci flow on Fano manifolds (I)

  • Xiuxiong Chen

    University of Wisconsin-Madison, Madison, USA
  • Bing Wang

    Princeton University, USA
The Kähler Ricci flow on Fano manifolds (I) cover
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Abstract

We study the evolution of pluri-anticanonical line bundles KMνK_M^{-\nu} along the Kähler Ricci flow on a Fano manifold MM. Under some special conditions, we show that the convergence of this flow is determined by the properties of the pluri-anticanonical divisors of MM. For example, the Kähler Ricci flow on MM converges when MM is a Fano surface satisfying c12(M)=1c_1^2(M)=1 or c12(M)=3c_1^2(M)=3. Combined with the works in [CW1] and [CW2], this gives a Ricci flow proof of the Calabi conjecture on Fano surfaces with reductive automorphism groups. The original proof of this conjecture is due to Gang Tian in [Tian90].

Cite this article

Xiuxiong Chen, Bing Wang, The Kähler Ricci flow on Fano manifolds (I). J. Eur. Math. Soc. 14 (2012), no. 6, pp. 2001–2038

DOI 10.4171/JEMS/353